A Model-Based Controller for Interactive Delayed Force Feedback Virtual Environments H. ARIOUI, A. KHEDDAR and S. MAMMAR Laboratoire Syst` emes Complexes, CNRS FRE-2494, 40, rue du Pelvoux, 91020 Evry Cedex, France; email: arioui,kheddar,mammar@iup.univ-evry.fr Received 2002, revised 2003 Abstract. This paper addresses the stability of time-delayed force-reflecting dis- plays used in virtual reality interactive systems. A novel predictive-like approach is proposed. The developed solution is stable and robust. Neither time delay esti- mation nor time delay behavior’s knowledge are required. The controller applies to constant or time-varying delays without any adaptation. In this research, efforts are devoted towards making results easy to implement in commercial haptic libraries and interface build-in controllers. Moreover, although this study focuses on virtual environments haptics, it can easily spreads to force feedback teleoperators. Keywords: virtual environment haptics, time-delayed force feedback, stability. 1. Introduction Virtual reality techniques refer typically to human-in-the-loop or hu- man centered advanced simulation or prototyping systems. The original feature of the concept lies in the multi-modality of the man-machine interaction involving all the human sensory modalities. Among these ca- pabilities, haptic feedback is an important issue allowing the human op- erator to experience manipulation and touching of virtual objects with realistic sensations of stiffness, roughness, temperature, shape, weight, contact forces, etc. In real worlds, these parameters are collected then interpreted by the human haptic sense through direct touch (contact). Virtual environments (VE) are visually rendered to the human oper- ator through computer screens, head mounted displays, workbenches, etc. To display 3D virtual sound simple headphones can be used. In the contrary to vision and auditory modalities, haptics requires active displays. Indeed, the used interfaces must be able to constraint human desired motions or to apply forces on the involved part (e.g. the human hand). Haptic displays are typically robotic-like devices which: (i) track hand motion (or applied forces) to be sent to the VE engine, and (ii) render subsequent VE reaction forces. Reaction forces are calculated thanks to computer haptics algorithms (collision detection, dynamic contact computation, etc.). There are many applications that use haptic feedback technology. Among the well known ones: interactive surgi- � 2003 Kluwer Academic Publishers. Printed in the Netherlands. c p.1
2 H. ARIOUI ET AL. cal simulators, interactive driving simulators, interactive games, VE based teleoperation and a great demand in industry virtual prototyp- ing. The last one would extend to concurrent engineering and would make available haptic interaction among a group of users sharing the same VE over a network. It is well known that the haptic loop requires a high bandwidth of 1 kHz, ideally, to guarantee the stability and the transparency of the haptic interaction. Data transmission time delay may compromise both stability and transparency of the haptic loop. Indeed the VE and the haptic interface may be distant from each other. This is the case of some applications such as concurrent engineering or shared VE. Moreover, heavy compu- tational haptics may cause time-delay in the simulation loop. There is few work addressing the stability of VE delayed haptics since, in most applications, the user is not distant from the interactive simulation. This papers proposes a simple and efficient solution to deal with this problem. As it will be discussed, the proposed solution holds for con- stant and time-varying delay without any change or adaptation of the controller. 2. Time-Delay Force Reflecting Controller VE haptic display controllers are mainly an adaptation of force reflect- ing teleoperators controllers. Nowadays, time delay is still known to be one of the most hard problems in force reflecting teleoperation. Many solutions, based on control theory, were proposed to deal with this prob- lem. Some of the most attractive ones are based on passivity. Anderson and Spong derived a stable controller from scattering network theory (Anderson, 1992). Their proposed controller guarantees stability under any communication time delay. Niemeyer and Slotine also proposed a passivity-based controller thanks to a “wave variable” formulation of the problem (Niemeyer, 1991; Niemeyer, 1997) also derived from network theory. A controller using geometric scattering and based on a Hamiltonian modelling of the teleoperator was proposed in (Stramigi- oli, 2000). In (Yokokojhi, 1999; Yokokojhi, 2000) and (Niemeyer, 1998), extensions and adaptations were made to deal with time-varying delay. Besides passivity based controllers, there are other methods, obviously less conservative, that have been investigated to deal with delayed force feedback. Indeed some controllers have been derived from the well known Lyapunov stability theory, see (Eusebi, 1998; Oboe, 1998), whereas in (Leung, 1995), a µ -synthesis approach where time-delay is modelled as a disturbance was proposed. In (Kosuge, 1996) a simple buffering technique to deal with time-varying delay was also developed. p.2
✁ � ✂ � ✄ ✂ 3 A MODEL-BASED CONTROLLER FOR TIME-DELAYED HAPTICS In the frame of VE haptic feedback, Hannaford et al. proposed a controller relaying on a “passivity observer” (Hannaford, 2002). The observer estimates the energy excess meanwhile, the controller cancels the estimated amount of energy so that force feedback is always passive. For delayed-process control, Smith prediction method is known since 1959 (Smith, 1959). However, it was not implemented in the frame of delayed force reflecting teleoperation. The reason that prohibits using Smith prediction lies in the practical impossibility to predict mainly (i) the remote environment behavior and, (ii) the operator desired trajectories, since they are given on-line. 2.1. Main result Before addressing the VE haptics context, let us recall some generic results that will be used in our proposed solution. Figure 1 illustrates any interconnected pair of passive systems defined respectively by their time-domain or frequency-domain linear mapping g 1 and g 2 . The exter- nal input signals of the interconnected systems are denoted respectively by e 1 and e 2 ∈ L 21 , the output signals by y 1 and y 2 ∈ L 2 , where as u 1 = e 1 − y 2 and u 2 = e 2 + y 1 are respectively the control signals. Eventually, the systems output signals y may be delayed by respectively τ 1 and τ 2 . e g _ + τ τ + e ✂ y ✂ + u g Figure 1. Interconnected systems with time delay. THEOREM 1 (Shaft, 1996). Considering τ 1 = τ 2 = 1 (identity map- ping which means no time-delay), if g 1 , g 2 are passive then the resulting system with inputs ( e 1 , e 2 ) ∈ L 2 and outputs ( y 1 , y 2 ) ∈ L 2 is also passive. Proof. see (Schaft, 1996; Vidyasagar, 1993) for the demonstration. 1 L 2 [0 , ∞ ) = L 2 consists of all functions f : R + → R which are measurable and satisfy � ∞ | f ( t ) | 2 dt < ∞ 0 p.3
4 H. ARIOUI ET AL. This passivity property is not preserved when there exist a trans- mission delay in the closed loop system, i.e. τ 1 ( t ) � = 0 or τ 2 ( t ) � = 0 Haptic device part y 1 + g 1 e 1 _ + + g 1 s 1 τ 2 τ 1 �������������������� + g 1 _ s 2 y 2 + g 2 e 2 + VE part Figure 2. Model-based controller. THEOREM 2. The interconnected delayed system shown in figure 1 can be stabilized, using a process-model based control of either g 1 or g 2 as illustrated in figure 2. Proof. Let f ∗ g denotes the convolution product of the functions f ( t ) and g ( t ): � ∞ f ∗ g = f ( t − ν ) g ( ν ) dν −∞ we recall that the convolution product is commutative. Let h ( τ i , t ) denotes the impulse response of the transmission channel (considered to be a delay-variable filter). The knowledge of this impulse response completely characterizes the transmission channel. In the case of none stationary channels (the case of variable delays), one can de- fine relations similar to stationary ones: for example the convolution product. In this case however, the impulse response h depends on two parameters: ( τ i and t ), which denotes the system output at instant τ i for an input at instant t . The convolution product can be generalized with the relation, known as Bello function (Bello, 1963): � ∞ z ( t ) = x ( t ) ∗ h ( τ, t ) = x ( t − τ ) h ( τ, t ) dτ (1) −∞ p.4
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